Soda-lime glass is a common engineering material thataccounts for over 90% of manufactured glasses. Superioroptical transparency, good scratch resistance, high stiffnessand hardness, very high compression strength, excellentheat, and chemical resistance are some of the many attrac-tive features that make glass popular for structural applica-tions besides its low cost, recyclability, sustainability, andarchitectural esthetics. Further, polymer-laminated glasses1
are found in many other applications, including automotivewindshields, windows, etc. Yet, this brittle material suffersfrom inadequacies such as low fracture toughness present-ing the possibility of catastrophic failure during service. Inapplications such as electronic displays and window panes
where stresses due to contact loading or those around pre-existing flaws drastically affect its performance.
Given its low fracture toughness, failure of glass is gener-ally abrupt and catastrophic in the presence of flaws, surfacescratches and/or cracks. Performing full-field, noncontactmeasurement of deformations and stresses to evaluatemechanical properties of glass could prove beneficial forimproving it’s performance as well as isolating/locating flawsin them which could otherwise be detrimental. For example,the surface flaws generated while handling glass may not bereadily visible to the naked eye but they significantly affectthe mechanical strength. Additional challenges such as extre-mely small and highly localized region of intense deforma-tions due to cracks, notches and other stress concentrationspose difficulties to measurement methods. Most reported
works on glasses2-4 have often relied on postmortem inspec-tion of fracture surfaces and/or crack growth morphologies.As glass is weakly birefringent relative to its strain at failure,it is difficult to obtain sufficient number of fringes to effec-tively analyze them photoelastically. An attempt in thisregard by Voloshin and Burger5 to analyze the stress field,using the so-called half- fringe photoelasticity is noteworthy.The method of caustics has also been used in the past to studycrack-tip and contact stress problems under quasi-static load-ing conditions.6,7
The lack of a suitable, easy to implement, full-field, non-contact optical technique to study glass (and transparentceramics in general) has been somewhat striking and needsattention. In this context, this work examines the feasibilityof a full-field optical technique called Digital GradientSensing (DGS)8 to tackle the stated problem. In DGS, theelasto-optic effects exhibited by transparent soda-lime glasssubjected to nonuniform state-of-stress are quantified byevaluating the angular deflections of light rays propagatingthrough the material. The angular deflection measurementsrepresent two orthogonal in-plane stress gradients underplane stress conditions. The presence of stress singularity atan impact point or a crack-tip makes DGS highly effectivenot only for mechanical field measurements but for locatingthe stress risers. Authors have previously reported variousstudies on transparent polymers9,10 demonstrating the effi-cacy of DGS. The current work is an attempt to extend it tostudy extremely brittle material such as soda-lime glass. Asnoted earlier, the angular deflections produced in glass areextremely small. Hence, the experimental parameters had tobe altered to tackle this challenge.
In this work, the working principle of DGS and its exten-sion to higher stiffness material such as soda-lime glass isdetailed first. The governing equation and its advantages aredetailed next. Two different experiments are reported in thisstudy to demonstrate the feasibility of the DGS for perform-ing accurate and reliable measurements in soda-lime glassunder quasi-loading conditions. First, stress concentrationdue to line-load acting on the edge of a planar sheet is stud-ied. The load histories are evaluated from the DGS measure-ments and compared with that obtained from a load-cell.Second, deformations near a notch-tip due to the appliedloads are studied. Stress intensity factors (SIFs) are evaluatedfrom experiments and examined relative to analytical coun-terparts. The results are summarized at the end of the report.
2 | DIGITAL GRADIENT SENSING(DGS)
2.1 | Experimental details
A schematic of the experimental setup for transmission-mode DGS technique is shown in Figure 1. In this
technique8, a planar surface spray painted with randomspeckles, henceforth referred to as “target,”is photographedthrough a planar transparent sheet being studied. Ordinarywhite light illumination is used for recording the grayscales on the target. The speckle pattern is first pho-tographed through the specimen in its undeformed state toobtain a reference image. That is, a point P on the targetplane (x0�y0 plane) is recorded by the camera throughpoint O on the specimen plane (x�y plane). Upon loading,the nonuniform stresses due to the imposed loads changethe refractive index of the specimen in the crack-tip vicin-ity. Additionally, the Poisson effect produces nonuniformthickness changes. A combination of these, commonlyreferred to as the elasto-optic effect, makes the light rays todeviate from their original path as they propagate in thevicinity of the stress riser such as a crack-tip. The specklepattern is once again photographed through the specimenin the deformed state. Now, a neighboring point Q on thetarget plane is recorded by the camera through point O onthe specimen plane after deformation. The local deviationsof light rays can be quantified by correlating speckleimages in the deformed and reference states to find speckledisplacements dx and dy. The angular deflections of lightrays /x and /y in two orthogonal planes (x�z and y�zplanes with the z-axis coinciding with the optical/cameraaxis of the setup and x�y being the specimen plane coordi-nates) can be computed if the distance between the speci-men plane and the target plane is known. A detailedanalysis under paraxial conditions shows that the localangular deflections are related to the gradients of in-planenormal stresses as,
/x;y ¼ � CrB@ðrx þ ryÞ@ðx; yÞ (1)
It is important to note that while recording the images, thecamera is focused on the “target” (and speckles) throughthe transparent sheet. Yet, the analysis requires mechanicalfields described on the specimen plane, situated at a dis-tance L from the target. That is, a point O(x, y) on thespecimen corresponds to a point P(x0, y0) on the targetplane as shown in the 2D schematic (see, Figure 2). Thiscan be accomplished using a pin-hole camera approxima-tion for which a mapping function between the specimenand the target planes can be deduced. In Figure 2,tan h ¼ ds
L ¼ dtLþD, where ds and dt are identified on the
specimen and target planes, respectively. Henceds ¼ ðL=ðLþ DÞÞdt. (A similar mapping function for thehorizontal plane is obvious and should be understood.)Glass being a very brittle, low toughness material, theresulting angular deflections due to applied load are gener-ally very small (a few micro-radians). Hence, D needs tobe increased significantly to accommodate for this, differ-ent from the assumption of D�L made by the authors’ in
SUNDARAM AND TIPPUR | 115
earlier investigations9-11 on polymers, and necessitatesaccounting for the perspective effect during analysis.
2.2 | Advantages of DGS
Transmission DGS is a full-field optical methodology thatexploits the technique of 2D digital image correlation(DIC) in conjunction with elasto-optic effect to measureangular deflections of light rays and thus stress gradients.A couple of advantages of the method relevant to thisresearch are (i) the singularity in the measured crack-tipfields facilitates locating the crack-tip more reliably (see9
for details), and (ii) the ability to easily tune (increase inthis case) the measurement sensitivity to deal with the highmaterial stiffness coupled with low toughness of glass thatresults in extremely small changes in thickness and
refractive index due to the prevailing state of stress. Thatis, it could be effective despite a low elasto-optic coeffi-cient of soda-lime glass (40-50 times lower in magnitudecompared to transparent polymers such as PMMA andPC6,11), low fracture toughness (~0.8 MPa√m vs 1.1 and2.3 MPa√m for PMMA and PC) and high crack speeds(~1500 m/s vs ~300 m/s for PMMA and PC). DGS has afew additional benefits that are noteworthy as well. InDGS, the specimen needs no speckle decoration. Instead, asingle planar surface (“target”) with the desired speckledecoration can be used repeatedly in multiple experimentswithout introducing variability to the gaging pattern. Thisfurther reduces experimental effort in terms of creatingmultiple specimens with speckle decoration of comparableif not identical characteristics. Another important advantageof DGS is in studying stress concentration problems. Being
Specimen
OF
Q
P
B
L
x
y
Camera
x0
y0
zSpeckle target
Crack
FIGURE 1 The schematic of theexperimental setup for transmission-modeDigital Gradient Sensing (DGS) techniqueto determine stress gradients in transparentsheets 8 [Color figure can be viewed atwileyonlinelibrary.com]
FIGURE 2 2D Schematic for mappingtarget plane coordinates to the specimenplane coordinates8
116 | SUNDARAM AND TIPPUR
a stress gradient measurement technique, mechanical fieldsfrom DGS, typically visualized as contours of orthogonalangular deflection components, tend to converge to thelocation of the stress concentration, say, a crack-tip. Thisgreatly assists while identifying the spatial location of thestress singularity in the field-of-view. The availability oforthogonal stress gradients as a rectangular array of datalends itself to an easy implementation of novel 2D integra-tion schemes to estimate stresses accurately via postpro-cessing of DGS measurements.12
3 | LINE-LOAD ON THE EDGE OF AGLASS SHEET
3.1 | Experimental details
A quasi-statically applied line-load on the edge of a largeplanar glass sheet was studied first. A 100975 mm2 rectan-gular specimen machined from a commercially procuredplate glass/soda-lime glass sheet (elastic modulus ~70 GPa,Poisson’s ratio 0.22 and Cr �0.027910�10 m2/N)6 ofthickness 4.65 mm was used. The schematic of the experi-mental setup is shown in Figure 3. The specimen wasplaced on a hardened steel platform and subjected to aline-load, using a cylindrical steel pin (diameter 5 mmattached to an Instron 4465 crosshead) in displacementcontrol mode (crosshead speed ~0.005 mm/s). A targetplate painted with random black and white speckles wasplaced 712 mm away from the specimen mid-plane. A cou-ple of heavy black dots (see, Figure 4) were marked on thespeckle plane to relate the image dimensions to the actualtarget and subsequently the specimen dimensions. A PointGrey Grasshopper3 digital camera with a 18-108 mm focal
length zoom lens (F# 5.6) was used to record specklesthrough the specimen in the load-point vicinity. The cam-era was situated at a distance (L) of approximately445 mm from the specimen plane.
A reference image of the target was recorded throughthe transparent specimen in the region-of-interest at a smallload (<5 N). As the load increased, speckle images wererecorded using time-lapse photography (15 frames per min-ute). Speckle images in the loading point vicinity corre-sponding to the undeformed and a select deformed(3000 N) state are shown in Figure 4. The recorded imagescorrespond to 63 mm958 mm region on the target platewhich translates to approximately 24 mm922 mm on thespecimen. It shows that due to deformation of the speci-men, the speckles appear smeared near the loading point,whereas they seem relatively unaffected far away. Besidesthis, very little information can be visualized from thesespeckle images. The digitized speckle images (118491100pixels) recorded at different load levels were correlatedwith the one corresponding to the reference condition,using image analysis software ARAMIS�. As describedpreviously, an array of in-plane speckle displacements onthe target plane (and hence the specimen plane) was evalu-ated and converted into angular deflections of light rays /x
and /y. A facet/sub-image size of 25925 pixels(1 pixel�53 lm on the target plane) with an overlap of 20pixels was used in the image analysis for extractingdisplacement components.
3.2 | Evaluation of load history
The equation for /x and /y in terms of the applied loadcan be described, using the classical Flamant solution,8
FIGURE 3 Schematic for quasi-static line loading on the edge of a soda-lime glass sheet used in DGS [Color figure can be viewed atwileyonlinelibrary.com]
SUNDARAM AND TIPPUR | 117
/x;y ¼ � Cr2Fp
cosð2hÞ; sinð2hÞ½ �r2
(2)
where Cr is the elasto-optic constant, F is the applied loadand (r, h) denote polar coordinates defined in Figure 3.The mapping function described in an earlier section wastaken into consideration during the analysis. Note thataccounting for rigid body motions may be necessary byenforcing boundary conditions of the problem to quantifythe contour levels for further analysis. This can be done byforcing symmetry and far-field conditions for this problemas in Ref.13 The same can also be accomplished by addingconstant terms to Equation 2 as,
/x;y ¼ � Cr2Fp
½cosð2hÞ; sinð2hÞ�r2
þ Cx;y (3)
where Cx;y denote constants which account for electronicnoise and rigid motions if any. In this experiment, F wasevaluated using over-deterministic regression analysis ofthe measured /x. Alternatively, /y can also be used toevaluate F. Discrete angular deflection values around theloading point in the region 1≤r/B≤2 along with an angularextent of �80°≤h≤�50°, �40°≤h≤40° and 50°≤h≤80°were used in the regression analysis. This ensured avoidingthe region of dominant stress triaxiality along the freeedges and regions adjacent to zones where angular deflec-tions are nearly zero. This also helped to deal with the lack
(A)
(B)
FIGURE 4 Speckle images in the (A) undeformed and (B)deformed states recorded through the soda-lime glass specimensubjected to line-load
(A)
(B)
FIGURE 5 Angular deflection contour plots (contourinterval=8910�6 rad) proportional to stress gradients of (rx+ry) inthe (A) x- and (B) y-directions for a soda-lime glass specimen underquasi-static line-load on its edge [Color figure can be viewed atwileyonlinelibrary.com]
118 | SUNDARAM AND TIPPUR
of data in the close vicinity of the loading point due tofinite (limited) numerical aperture of the camera.
3.3 | Experimental results
The measured DGS contours (on the specimen plane) ofangular deflections /x and /y for a select load of 3000 Nare shown in Figure 5. It should be noted that the contoursnear the free edge of the specimen show unavoidable edgeeffects. Using Equation 3, the load was extracted at eachtime step to obtain the load history. By matching the timestamp on the recorded image with the data from the load-cell, another set of load history was obtained. Both the loadhistories are plotted in Figure 6 and the two are in goodagreement with each other.
4 | CRACK-TIP DEFORMATIONS
4.1 | Experimental details
Quasi-static crack-tip deformations were measured next. Asoda-lime glass sheet of 4.65 mm thickness was cut toobtain 130 mm950 mm SENB specimens. A 10 mm longnotch was machined into it, using a band saw of 2 mmthickness. The initial notch was intentionally kept wide toavoid premature specimen failure at lower load. The speci-men was placed on two anvils (120 mm apart) symmetri-cally relative to the notch cut along the bottom edge andthe loading pin pressing down on the top edge. A sym-metric 3-point-bend test was performed on this specimen,using Instron 4465 testing machine in displacement
FIGURE 6 Load history obtained from(A) DGS data and (B) load-cell of thetesting machine [Color figure can beviewed at wileyonlinelibrary.com]
FIGURE 7 Loading configuration forquasi-static 3-point bending of an edgenotched soda-lime glass plate [Color figurecan be viewed at wileyonlinelibrary.com]
SUNDARAM AND TIPPUR | 119
(A)
(B)
FIGURE 8 Speckle images in the (A) undeformed and (B)deformed states near the initial notch-tip recorded by the camerathrough the soda-lime glass specimen. (The notch is not in focus)
(A)
(B)
FIGURE 9 Angular deflection contour plots (contourinterval=8910�6 rad) proportional to stress gradients of (rx+ry) inthe (A) x- and (B) y-directions near the notch-tip for a soda-limeglass specimen under quasi-static 3-point bending load [Color figurecan be viewed at wileyonlinelibrary.com]
FIGURE 10 Comparison of mode-Istress intensity factors from DGS data andthe analytical expression (Equation 6)[Color figure can be viewed atwileyonlinelibrary.com]
120 | SUNDARAM AND TIPPUR
control mode (crosshead speed of ~0.005 mm/s). Theschematic of the experimental setup used is shown in Fig-ure 7.
The speckled target plate was placed behind the specimenat a distance of D ~712 mm from the specimen mid-plane.A Point Grey Grasshopper3 digital camera (spatial resolutionof 118491100 pixels and 10-bit gray scale resolution) fittedwith a macro zoom lens of 18-108 mm focal length (F# 5.6)was used to record speckles on the target plate as in the pre-vious experiment. The camera was placed in front of thespecimen at a distance (L) of ~445 mm with the camerafocused on the target plate through the specimen. Two CFLlamps were used to illuminate the target plate uniformly.
A reference speckle image of the target plate wasrecorded through the crack-tip vicinity at no-load(load<5 N) condition. As the load was increased gradually,the perturbed images of the speckles on the target platewere recorded, using time-lapse photography at 15 framesper minute. Two representative images, one in the refer-ence state and the other in the deformed state, are shownin Figure 8. The recorded images correspond to63 mm958 mm region on the target plate which translatesto approximately 24 mm922 mm on the specimen. Usinga pair of reference dots (see, Figure 8) marked on the tar-get plate, the dimensions on the image in terms of pixelswere related back to target plate dimensions (1 pixel=approx. 53 lm on the target plane) and then to specimendimensions (1 pixel=approx. 21 lm on the specimenplane). Sufficient care was also exercised to obtain a nearGaussian distribution of gray scales for each image in themid-range of the gray scale by positioning the lampsappropriately. When looked carefully, the speckles appearslightly smeared in Figure 8B around the crack-tip whereasthey seem largely unaffected away from it. The imageswere again correlated, using ARAMIS� software (GoMGmbH, Braunschweig, Germany). A facet/sub-image sizeof 25925 pixels with an overlap of 20 pixels (ie, step size
of 5 pixels) was used during speckle correlation. Theresulting data matrix was exported to MATLAB for post-processing, including the evaluation of orthogonal angulardeflections in the region of interest.
4.2 | Evaluation of stress intensity factor(SIF)
Williams’ asymptotic stress fields for angular deflectionsnear a statically loaded mode-I crack-tip become,9,14
/x ¼ CrB@ðrx þ ryÞ
@x
¼ CrBX1N¼1
ANN2� 1
� �r
N2�2ð Þcos N
2� 2
� �h
� � (4)
/y ¼� CrB@ðrx þ ryÞ
@y
¼ �CrBX1N¼1
ANN2� 1
� �r
N2�2ð Þsin N
2� 2
� �h
� �
(5)
where (r,h) denote the crack-tip polar coordinates,A1 ¼ KI
ffiffiffiffiffiffiffiffi2=p
pwith KI being the mode-I stress intensity
factor. (It should be noted that the dominant or the lead-ing term of the crack-tip stress fields for ðrx þ ryÞremain unaffected in the classical Creager’s solution15
that takes into account the finite root radius of a notch.)In this experiment, the SIFs were evaluated using Equa-tion 4 by employing an over-deterministic regressionanalysis of the measured data and N=4. Discrete angulardeflection values around the crack-tip in the region0.5≤r/B≤1.5 along with the angular extent of�150°≤h≤+150°, was used in the regression analysis.This ensured that the data used was sufficiently close tothe crack-tip yet outside the region of significant stresstriaxiality. This also indirectly helped to minimize theerror while locating the crack-tip due to potential edgeeffects during image correlation. It should be noted thatthe mapping function, described in an earlier section,was accounted for during the analysis. Error bars wereobtained as a result of uncertainty in locating the crack-tip and by using various subsets of the region describedabove.
The mode-I SIFs can also be evaluated from the load his-tory obtained from the load-cell and the sample geometry as,9
where F is the applied load, S is the span, B is the thick-ness of the specimen, w is the width of the specimen and ais the initial crack length.
4.3 | Experimental results
The angular deflection contours on the specimen plane intwo in-plane orthogonal directions at a load of 880 N areshown in Figure 9. A heavy white line is overlaid on the
KI ¼ F SBw3=2
3ðnÞ1=2 1:99� nð1� nÞ 2:15� 3:93ðnÞ þ 2:7ðnÞ2n oh i
2ð1þ 2nÞð1� nÞ3=2; n ¼ a
w(6)
SUNDARAM AND TIPPUR | 121
resulting contours to represent the notch. The mode-I SIFsobtained from the regression analysis of measured data atdifferent imposed load levels is plotted in Figure 10 alongwith that obtained from the load-cell. Evidently, there is agood agreement between the two.
5 | CONCLUSIONS
The transmission-mode Digital Gradient Sensing (DGS)method has been extended in this work to visualize and quan-tify orthogonal stress gradient fields in soda-lime glass underquasi-static loading conditions. Typical challenges of study-ing failure of high stiffness and low toughness transparentceramics have been overcome by using this full-field speckle-based optical method. The feasibility of DGS to map stressgradient fields due to quasi-statically applied line-load on theedge of a uniformly supported plate and on a symmetricallybent beam with a single edge notch is demonstrated. The full-field measurements have been analyzed to extract the appliedload and mode-I stress intensity factor, respectively, in theseconfigurations with good accuracy. Thus, demonstrating theaccuracy and robustness of the measurement technique andthe methodology of analysis for soda-lime glass.
ACKNOWLEDGMENTS
The support of the U.S. Army Research Office for thisresearch through grants W911NF-16-1-0093 and W911NF-15-1-0357 (DURIP) are gratefully acknowledged.
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How to cite this article: Sundaram BM, Tippur HV.Full-field measurement of contact-point and crack-tipdeformations in soda-lime glass. Part-I: Quasi-staticLoading. Int J Appl Glass Sci. 2018;9:114-122.https://doi.org/10.1111/ijag.12278
